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Theorem br2ndeq 31647
Description: Uniqueness condition for binary relationship over the 2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br2ndeq.1 𝐴 ∈ V
br2ndeq.2 𝐵 ∈ V
br2ndeq.3 𝐶 ∈ V
Assertion
Ref Expression
br2ndeq (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)

Proof of Theorem br2ndeq
StepHypRef Expression
1 br2ndeq.1 . . . 4 𝐴 ∈ V
2 br2ndeq.2 . . . 4 𝐵 ∈ V
31, 2op2nd 7162 . . 3 (2nd ‘⟨𝐴, 𝐵⟩) = 𝐵
43eqeq1i 2625 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶𝐵 = 𝐶)
5 fo2nd 7174 . . . 4 2nd :V–onto→V
6 fofn 6104 . . . 4 (2nd :V–onto→V → 2nd Fn V)
75, 6ax-mp 5 . . 3 2nd Fn V
8 opex 4923 . . 3 𝐴, 𝐵⟩ ∈ V
9 fnbrfvb 6223 . . 3 ((2nd Fn V ∧ ⟨𝐴, 𝐵⟩ ∈ V) → ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶))
107, 8, 9mp2an 707 . 2 ((2nd ‘⟨𝐴, 𝐵⟩) = 𝐶 ↔ ⟨𝐴, 𝐵⟩2nd 𝐶)
11 eqcom 2627 . 2 (𝐵 = 𝐶𝐶 = 𝐵)
124, 10, 113bitr3i 290 1 (⟨𝐴, 𝐵⟩2nd 𝐶𝐶 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174   class class class wbr 4644   Fn wfn 5871  ontowfo 5874  cfv 5876  2nd c2nd 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fo 5882  df-fv 5884  df-2nd 7154
This theorem is referenced by:  br2ndeqg  31649  dfrn5  31651  brtxp  31962  brpprod  31967  elfuns  31997  brimg  32019  brcup  32021  brcap  32022  brrestrict  32031
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